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Tuesday, October 30, 2007

This and That

I'm still kind of stressed out and don't come around to finishing a couple of posts that have been stuck in draft stage for a while. I am having fun answering emails the whole day from workshop speakers who want to know whether there will be chalk in the seminar room, or are worried whether they will get dinner in Waterloo after 10pm (a justified worry btw). No, I like that, I'm serious, makes me feel so important. Either way, some interesting things I came across lately:

Sorry, but I found the "turning the sphere inside out" thing deeply *un*interesting. You are allowed to make it *pass through itself* but you are forbidden sharp corners? Why? So that you can prove something. This is why mathematics can never be as interesting as physics.....

Beowulf,I think maybe its because if you embed a sphere in more dimensions, the new allowable motions, when projected would look like you were passing it through itself. So if you are studying intrinsic properties of the sphere it makes sense to allow it to pass through itself (as in the Klein bottle construction) but not to make edges (which doesnt correspond to allowable motions in more dimensions).

I can't stand the Starbucks sandwiches! I think tuna is the worst. No wait, actually they have one with dried tomatoes which I find even worse. But I like their carrot cake :-) What I will never understand though is why they don't have spoons in the Starbucks in North America (they do have spoons in Europe). I never know what to do with the milk foam. I've seen customers trying to use the wooden sticks, but of course it doesn't really work. Best,

well, at least where I've been lately, let me see. Germany, Austria, France... oohm, they don't have Starbucks in Italy apparently, neither in Poland, and I couldn't find one in Lisboa... okay, apologies, I admit it's not exactly a representative sample. Best - B.

tthis mathematical trick is very surprising, and the movie is really impressive. And, it is more than 10 years old, which makes the tricks even more stunning...

Hi beowulf,

I agree that in the movie, the requirement that all allowed transformations have to avoid kinks appears quite ad hoc, and arbitrary.

However, there is a precise mathematical definition of this "non-kinkiness" which is very reasonable: the requirement is that the deformations are continous, or even smooth, and that the shape which is transformed keeps the structure of a manifold all the time, which means that for each point on the shape there is a small neighbourhood which "looks like" the plane, or more generally like a neighbourhood in n-dimensional Euclidian space.

The restriction on possible transformations implied by this definition has even some physical equivalent: if you imagine the shape as, say, a freely floating soap film, bending and stretching of the film is allowed - and the film may have a chance to sustain these kinds of deformations. However, the kink will always and necessarily introduce some singular point, where the film cannot avoid rupture (or it has to be kept fixed from the outside).

Of course, there is still this tricky problem that mathematicians sometimes don't care about that real soap films always have to be embedded in three-space, and that the "pass through itself" may not work in practice ;-)

Hi Stefan, I agree that it is reasonable to require the sphere to remain a manifold. That is natural. It is even more natural, however, to forbid self-intersections! Immersions/embeddings. The problem then, of course, is that you get a boring theorem which is easy to prove....yes, as pedagogy it is very impressive; but as mathematics......

I wish there *were* Starbucks in Italy. Italians could go there to find out how unbelievably *bad* coffee is everywhere outside Italy, and feel more grateful for being Italians. On the other hand, I have actually seen Italians going to McDonald's. To get there, they had to walk past a family-run trattoria serving food better than anything you will find on the entire North American continent.